Continual Learning of Nonlinear Independent Representations (2408.05788v1)

Abstract: Identifying the causal relations between interested variables plays a pivotal role in representation learning as it provides deep insights into the dataset. Identifiability, as the central theme of this approach, normally hinges on leveraging data from multiple distributions (intervention, distribution shift, time series, etc.). Despite the exciting development in this field, a practical but often overlooked problem is: what if those distribution shifts happen sequentially? In contrast, any intelligence possesses the capacity to abstract and refine learned knowledge sequentially -- lifelong learning. In this paper, with a particular focus on the nonlinear independent component analysis (ICA) framework, we move one step forward toward the question of enabling models to learn meaningful (identifiable) representations in a sequential manner, termed continual causal representation learning. We theoretically demonstrate that model identifiability progresses from a subspace level to a component-wise level as the number of distributions increases. Empirically, we show that our method achieves performance comparable to nonlinear ICA methods trained jointly on multiple offline distributions and, surprisingly, the incoming new distribution does not necessarily benefit the identification of all latent variables.

• The paper introduces a novel approach that improves nonlinear ICA identifiability through continual learning across sequential distributions.
• It employs a combination of VAEs and GEM to reconstruct observations and preserve previously learned representations while mitigating catastrophic forgetting.
• Theoretical analysis shows component-wise identification is achievable with 2n+1 distributions, highlighting its practical impact on adaptive systems.

Continual Learning of Nonlinear Independent Representations

Introduction

The paper "Continual Learning of Nonlinear Independent Representations" addresses the challenge of learning identifiable representations from sequentially arriving distributions. This work explores continual causal representation learning (CCRL), with a focus on the nonlinear Independent Component Analysis (ICA) framework. The principal aim is to examine how model identification improves with each additional distribution, progressing from subspace-level to component-wise identifiability.

Theoretical Insights

The authors investigate the identifiability of latent variables within the context of nonlinear ICA. The paper asserts that identifiability increases with the number of distributions, achieving component-wise identification with $2n+1$ distributions and subspace identification with $n+1$ distributions. This is grounded on several foundational assumptions, such as the independence of latent variables conditioned on the domain and the smoothness of conditional densities. Theoretical results suggest that while identifiability is not always guaranteed without further assumptions, leveraging multiple distributions can enhance it significantly.

Two primary outcomes are highlighted:

1. Component-wise Identifiability: Demonstrated to be achievable with $2n+1$ distributions if certain matrix invertibility conditions are met.
2. Subspace Identifiability: Attainable with $n+1$ distributions, where true latent variables can be expressed as a subspace of estimated ones.

Methodology

The paper presents an approach to CCRL that employs Variational Autoencoders (VAEs) paired with Gradient Episodic Memory (GEM) to ensure that the learning of new distributions does not degrade the model's performance on previous ones. The key objectives include:

• Reconstruction of observations within the current distribution.
• Preservation of the representation learned from earlier distributions.

The use of GEM is critical as it constrains the gradient updates by aligning the gradients of new and old distributions, thus preventing catastrophic forgetting. This alignment is achieved through solving a quadratic programming problem to find an optimal gradient direction that minimizes loss on new data while maintaining performance on previous distributions.

Empirical Evaluation

The authors conduct extensive experiments using synthetic datasets, where latent variables follow predefined distributions. The results indicate that the continual learning approach performs comparably to joint training methods, which require simultaneous access to all distributions. Notably, the authors find that as more distributions are observed, the model's ability to identify latent variables improves, validating their theoretical claims.

Moreover, the paper demonstrates that in certain scenarios, incremental addition of distributions can impair the identifiability of some latent variables. This phenomenon is attributed to the fact that new distributions can introduce noise that affects the model's ability to correctly identify latent variables. The continual learning setup, however, offers flexibility to maintain previously learned knowledge, thus often leading to better identifiability for specific variables.

Implications and Future Directions

The practical implications of this work extend to real-world applications where data is collected sequentially, such as in autonomous systems, medical diagnosis, and adaptive learning platforms. By enabling models to learn causal representations continually, this approach mirrors human learning more closely, where knowledge is incrementally updated over time.

From a theoretical standpoint, this work contributes to the understanding of how sequential distribution shifts can aid in learning identifiable representations. However, the requirement of knowing the number of changing variables in advance remains a significant limitation. Future research could focus on developing methods to dynamically determine the number of changing variables within a continual learning framework.

Conclusion

This paper makes significant strides in advancing the field of continual causal representation learning. By theoretically and empirically demonstrating that identifiability improves with more distributions, it provides a robust framework for learning in sequential contexts. The combination of VAEs and GEM presents a viable solution to address the challenges posed by continual learning, ensuring that models can adapt to incoming data without compromising previously acquired knowledge. This work paves the way for future explorations into more complex causal representation learning tasks, ultimately bridging the gap between theoretical insights and practical applications.